Equilateral and Isosceles Triangles - Mrs. Barnhart's Classes

5.4 Equilateral and Isosceles Triangles

Essential Question What conjectures can you make about the side

lengths and angle measures of an isosceles triangle?

CONSTRUCTING VIABLE ARGUMENTS

To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Writing a Conjecture about Isosceles Triangles

Work with a partner. Use dynamic geometry software. a. Construct a circle with a radius of 3 units centered at the origin. b. Construct ABC so that B and C are on the circle and A is at the origin.

C

3

2

1

0

-4

-3

-2

-1 A 0

1

-1

-2

-3

B

2

3

4

Sample

Points A(0, 0) B(2.64, 1.42) C(-1.42, 2.64) Segments AB = 3 AC = 3 BC = 4.24 Angles mA = 90? mB = 45? mC = 45?

c. Recall that a triangle is isosceles if it has at least two congruent sides. Explain why ABC is an isosceles triangle.

d. What do you observe about the angles of ABC?

e. Repeat parts (a)?(d) with several other isosceles triangles using circles of different radii. Keep track of your observations by copying and completing the table below. Then write a conjecture about the angle measures of an isosceles triangle.

A

B

C

AB AC BC mA mB mC

Sample 1. (0, 0) (2.64, 1.42) (-1.42, 2.64) 3 3 4.24 90? 45? 45?

2. (0, 0)

3. (0, 0)

4. (0, 0)

5. (0, 0)

f. Write the converse of the conjecture you wrote in part (e). Is the converse true?

Communicate Your Answer

2. What conjectures can you make about the side lengths and angle measures of an isosceles triangle?

3. How would you prove your conclusion in Exploration 1(e)? in Exploration 1(f)?

Section 5.4 Equilateral and Isosceles Triangles 251

5.4 Lesson

Core Vocabulary

legs, p. 252 vertex angle, p. 252 base, p. 252 base angles, p. 252

What You Will Learn

Use the Base Angles Theorem. Use isosceles and equilateral triangles.

Using the Base Angles Theorem

A triangle is isosceles when it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs. The angle formed by the legs is the vertex angle. The third side is the base of the isosceles triangle. The two angles adjacent to the base are called base angles.

Theorems

Theorem 5.6 Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent.

If A--B A--C, then B C.

Proof p. 252

vertex angle

leg

leg

base angles

base

A

B

C

Theorem 5.7 Converse of the Base Angles Theorem

If two angles of a triangle are congruent, then the sides

A

opposite them are congruent.

If B C, then A--B A--C.

Proof Ex. 27, p. 275

B

C

Base Angles Theorem

B

Given A--B A--C

Prove B C

A

D

Plan a. Draw A--D so that it bisects CAB.

C

for Proof

b. Use the SAS Congruence Theorem to show that ADB ADC.

c. Use properties of congruent triangles to show that B C.

Plan STATEMENTS

in Action

a.

1.

Draw A--D, the angle

bisector of CAB.

REASONS 1. Construction of angle bisector

2. CAD BAD

3. A--B A--C 4. D--A D--A

2. Definition of angle bisector 3. Given 4. Reflexive Property of Congruence (Thm. 2.1)

b. 5. ADB ADC 5. SAS Congruence Theorem (Thm. 5.5)

c. 6. B C

6. Corresponding parts of congruent triangles are congruent.

252 Chapter 5 Congruent Triangles

Using the Base Angles Theorem

In DEF, D--E D--F. Name two congruent angles.

F

E

D

SOLUTION

D--E D--F, so by the Base Angles Theorem, E F.

Monitoring Progress

Help in English and Spanish at

Copy and complete the statement.

1. If H--G H--K, then

.

H

2. If KHJ KJH, then

.

G

K

J

Recall that an equilateral triangle has three congruent sides.

READING

The corollaries state that a triangle is equilateral if and only if it is equiangular.

Corollaries

Corollary 5.2 Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular.

A Proof Ex. 37, p. 258; Ex. 10, p. 353

Corollary 5.3 Corollary to the Converse of the Base Angles Theorem

If a triangle is equiangular, then it is equilateral.

B

C

Proof Ex. 39, p. 258

S

T

5 U

Finding Measures in a Triangle

Find the measures of P, Q, and R.

P

SOLUTION

The diagram shows that PQR is equilateral. So, by

R

the Corollary to the Base Angles Theorem, PQR is

equiangular. So, mP = mQ = mR.

3(mP) = 180?

Q Triangle Sum Theorem (Theorem 5.1)

mP = 60?

Divide each side by 3.

The measures of P, Q, and R are all 60?.

Monitoring Progress

Help in English and Spanish at

3. Find the length of S--T for the triangle at the left.

Section 5.4 Equilateral and Isosceles Triangles 253

Step 1

Using Isosceles and Equilateral Triangles

Constructing an Equilateral Triangle

Construct an equilateral triangle that has side lengths congruent to A--B. Use a

compass and straightedge.

SOLUTION Step 2

A

Step 3 C

B

Step 4 C

A

B

A

B

Copy a segment Copy A--B.

Draw an arc Draw an arc with center A and radius AB.

A

B

A

B

Draw an arc Draw an arc with center B and radius AB. Label the intersection of the arcs from Steps 2 and 3 as C.

AAsD--c--aiCBrrAmcaBaalewenrC,ecdAa.--irrB--BBactCdrleeiicai,aaAno--rAu--egfBsClerteha.A--deBDBiB--seiraCcoaamanfw.udetBsheye

the Transitive Property of

AC--CongrBu--eCn.cSeo(,TheAoBreCmis2.1),

equilateral.

COMMON ERROR

You cannot use N to refer to LNM because three angles have N as their vertex.

Using Isosceles and Equilateral Triangles

Find the values of x and y in the diagram.

K

4

y

L x + 1

N

M

SOLUTION

Step 1 Step 2

KF--iNndthK--eLv.aSluoe,

of y. Because y = 4.

KLN

is

equiangular,

it

is

also

equilateral

and

Find the value of x. Because LNM LMN, L--N L--M, and LMN is

isosceles. You also know that LN = 4 because KLN is equilateral.

LN = LM

Definition of congruent segments

4 = x + 1

Substitute 4 for LN and x + 1 for LM.

3 = x

Subtract 1 from each side.

254 Chapter 5 Congruent Triangles

Solving a Multi-Step Problem

In the lifeguard tower, P--S Q--R and QPS PQR.

P

Q

12

T

3

4

S

R

COMMON ERROR

When you redraw the triangles so that they do not overlap, be careful to copy all given information and labels correctly.

a. Explain how to prove that QPS PQR. b. Explain why PQT is isosceles.

SOLUTION

a.

PD--Qraw

aQ--nPd,lPa--bSel Q--QRP, SanadndQPPQS Rso PthQatRt.hSeoy,dboyntohte

overlap. You can see that SAS Congruence Theorem

(Theorem 5.5), QPS PQR.

P

Q

P

Q

2

1

T 3

T 4

S

R

b. tFrriaonmglpeasrat r(ea)c,oynogurukennotw. Bthyatthe 1Conver2sebeocfatuhseeBcaosrereAspnognledsinTghpeaorrtesmo,fP--cTongrQu--eTn,t

and PQT is isosceles.

Monitoring Progress

Help in English and Spanish at

4. Find the values of x and y in the diagram.

y? x?

5. In Example 4, show that PTS QTR.

Section 5.4 Equilateral and Isosceles Triangles 255

5.4 Exercises

Dynamic Solutions available at

Vocabulary and Core Concept Check

1. VOCABULARY Describe how to identify the vertex angle of an isosceles triangle.

2. WRITING What is the relationship between the base angles of an isosceles triangle? Explain.

Monitoring Progress and Modeling with Mathematics

In Exercises 3? 6, copy and complete the statement. State which theorem you used. (See Example 1.)

E

12. MODELING WITH MATHEMATICS A logo in an advertisement is an equilateral triangle with a side length of 7 centimeters. Sketch the logo and give the measure of each side.

A

B

C

D

3. If A--E D--E, then ___ ___.

4. If A--B E--B, then ___ ___.

5. If D CED, then ___ ___.

6. If EBC ECB, then ___ ___.

In Exercises 7?10, find the value of x. (See Example 2.)

7.

A

8.

M

x

12

x

60? 60?

B

C

L

16

N

9.

S

10. E 5

x?

R

T

5

3x? F

5 D

11. MODELING WITH MATHEMATICS The dimensions of a sports pennant are given in the diagram. Find the values of x and y.

79?

WC

y?

x?

In Exercises 13?16, find the values of x and y. (See Example 3.)

13.

y?

x?

14. y? 40? x?

15.

40

8y

40

x?

16.

3x - 5

5y - 4

y + 12

CONSTRUCTION In Exercises 17 and 18, construct an equilateral triangle whose sides are the given length.

17. 3 inches

18. 1.25 inches

19.

ERROR finding

ANALYSIS the length

oDf Be--sCcr.ibe

and

correct

the

error

in

B 5

A

6

Because A C,

A--C B--C.

So, BC = 6. C

256 Chapter 5 Congruent Triangles

20. PROBLEM SOLVING The diagram represents part of the exterior of the Bow Tower in Calgary, Alberta, Canada. In the diagram, ABD and CBD are congruent equilateral triangles. (See Example 4.)

a. Explain why ABC is isosceles.

b. Explain why BAE BCE.

c. Show that ABE and CBE are congruent.

d. Find the measure of BAE.

A BE D

C

21. FINDING A PATTERN In the pattern shown, each

small triangle is an equilateral triangle with an area

of 1 square unit.

a. Explain how you

Triangle

Area

know that any

1 square

triangle made

unit

out of equilateral

triangles is

equilateral.

b. Find the areas of the first four triangles in the pattern.

c. Describe any patterns in the areas. Predict the area of the seventh triangle in the pattern. Explain your reasoning.

22. REASONING The base of isosceles XYZ is Y--Z. What

can you prove? Select all that apply.

A X--Y X--Z

C Y Z

B X Y

D Y--Z Z--X

In Exercises 23 and 24, find the perimeter of the triangle.

23.

24. (21 - x) in.

7 in.

(x + 4) in.

(4x + 1) in.

(2x - 3) in. (x + 5) in.

MODELING WITH MATHEMATICS In Exercises 25?28, use the diagram based on the color wheel. The 12 triangles in the diagram are isosceles triangles with congruent vertex angles.

yellow- yellow yellow-

green

orange

green

orange

bluegreen

redorange

blue

red

blue-

red-

purple purple purple

25. Complementary colors lie directly opposite each other on the color wheel. Explain how you know that the yellow triangle is congruent to the purple triangle.

26. The measure of the vertex angle of the yellow triangle is 30?. Find the measures of the base angles.

27. Trace the color wheel. Then form a triangle whose vertices are the midpoints of the bases of the red, yellow, and blue triangles. (These colors are the primary colors.) What type of triangle is this?

28. Other triangles can be formed on the color wheel that are congruent to the triangle in Exercise 27. The colors on the vertices of these triangles are called triads. What are the possible triads?

29. CRITICAL THINKING Are isosceles triangles always acute triangles? Explain your reasoning.

30. CRITICAL THINKING Is it possible for an equilateral triangle to have an angle measure other than 60?? Explain your reasoning.

31. MATHEMATICAL CONNECTIONS The lengths of the sides of a triangle are 3t, 5t - 12, and t + 20. Find the values of t that make the triangle isosceles. Explain your reasoning.

32. MATHEMATICAL CONNECTIONS The measure of an exterior angle of an isosceles triangle is x?. Write expressions representing the possible angle measures of the triangle in terms of x.

33. WRITING Explain why the measure of the vertex angle of an isosceles triangle must be an even number of degrees when the measures of all the angles of the triangle are whole numbers.

Section 5.4 Equilateral and Isosceles Triangles 257

34. PROBLEM SOLVING The triangular faces of the peaks on a roof are congruent isosceles triangles with vertex angles U and V.

37. PROVING A COROLLARY Prove that the Corollary to the Base Angles Theorem (Corollary 5.2) follows from the Base Angles Theorem (Theorem 5.6).

U

V

6.5 m

W

8 m

X

Y

a. Name two angles congruent to WUX. Explain your reasoning.

b. Find the distance between points U and V.

35. PROBLEM SOLVING A boat is traveling parallel to

the shore along RT. When the boat is at point R, the

captain measures the angle to the lighthouse as 35?. After the boat has traveled 2.1 miles, the captain measures the angle to the lighthouse to be 70?.

R

2.1 mi S

T

35?

70?

L

a. Find SL. Explain your reasoning. b. Explain how to find the distance between the boat

and the shoreline.

36. THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer.

38. HOW DO YOU SEE IT? You are designing fabric purses to sell at the school fair.

B

C

100?

E

A

D

a. Explain why ABE DCE. b. Name the isosceles triangles in the purse. c. Name three angles that are congruent to EAD.

39. PROVING A COROLLARY Prove that the Corollary to the Converse of the Base Angles Theorem (Corollary 5.3) follows from the Converse of the Base Angles Theorem (Theorem 5.7).

40. MAKING AN ARGUMENT The coordinates of two points are T(0, 6) and U(6, 0). Your friend claims that points T, U, and V will always be the vertices of an isosceles triangle when V is any point on the line y = x. Is your friend correct? Explain your reasoning.

41. PROOF Use the diagram to prove that DEF is equilateral.

A

D

F E

B

C

Given ABC is equilateral. CAD ABE BCF

Prove DEF is equilateral.

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Use the given property to complete the statement. (Section 2.5)

42. Reflexive Property of Congruence (Theorem 2.1): ____ S--E 43. Symmetric Property of Congruence (Theorem 2.1): If ____ ____, then R--S J--K. 44. Transitive Property of Congruence (Theorem 2.1): If E--F P--Q, and P--Q U--V, then ____ ____.

258 Chapter 5 Congruent Triangles

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